Average Error: 34.9 → 10.1
Time: 16.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.322469259707235906288389821512289579342 \cdot 10^{143}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.116683443156571950895722510109441206323 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.322469259707235906288389821512289579342 \cdot 10^{143}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 3.116683443156571950895722510109441206323 \cdot 10^{-33}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} - \frac{b}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r61938 = b;
        double r61939 = -r61938;
        double r61940 = r61938 * r61938;
        double r61941 = 4.0;
        double r61942 = a;
        double r61943 = c;
        double r61944 = r61942 * r61943;
        double r61945 = r61941 * r61944;
        double r61946 = r61940 - r61945;
        double r61947 = sqrt(r61946);
        double r61948 = r61939 + r61947;
        double r61949 = 2.0;
        double r61950 = r61949 * r61942;
        double r61951 = r61948 / r61950;
        return r61951;
}
↓
double f(double a, double b, double c) {
        double r61952 = b;
        double r61953 = -2.322469259707236e+143;
        bool r61954 = r61952 <= r61953;
        double r61955 = 1.0;
        double r61956 = c;
        double r61957 = r61956 / r61952;
        double r61958 = a;
        double r61959 = r61952 / r61958;
        double r61960 = r61957 - r61959;
        double r61961 = r61955 * r61960;
        double r61962 = 3.116683443156572e-33;
        bool r61963 = r61952 <= r61962;
        double r61964 = r61952 * r61952;
        double r61965 = 4.0;
        double r61966 = r61958 * r61956;
        double r61967 = r61965 * r61966;
        double r61968 = r61964 - r61967;
        double r61969 = sqrt(r61968);
        double r61970 = 2.0;
        double r61971 = r61970 * r61958;
        double r61972 = r61969 / r61971;
        double r61973 = r61952 / r61971;
        double r61974 = r61972 - r61973;
        double r61975 = -1.0;
        double r61976 = r61975 * r61957;
        double r61977 = r61963 ? r61974 : r61976;
        double r61978 = r61954 ? r61961 : r61977;
        return r61978;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.9
Target21.3
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.322469259707236e+143

    1. Initial program 59.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Simplified59.5

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}}\]

    3. Taylor expanded around -inf 2.5

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

    4. Simplified2.5

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]

    if -2.322469259707236e+143 < b < 3.116683443156572e-33

    1. Initial program 14.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Simplified14.4

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}}\]
    3. Using strategy rm

    4. Applied div-sub14.5

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} - \frac{b}{2 \cdot a}}\]

    if 3.116683443156572e-33 < b

    1. Initial program 55.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Simplified55.5

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}}\]

    3. Taylor expanded around inf 6.5

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.322469259707235906288389821512289579342 \cdot 10^{143}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.116683443156571950895722510109441206323 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))